Integrand size = 19, antiderivative size = 411 \[ \int \sqrt {a+b x} \sqrt [6]{c+d x} \, dx=\frac {3 (b c-a d) \sqrt {a+b x} \sqrt [6]{c+d x}}{20 b d}+\frac {3 (a+b x)^{3/2} \sqrt [6]{c+d x}}{5 b}-\frac {3\ 3^{3/4} (b c-a d)^{5/3} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{b c-a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{40 b d^2 \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}} \]
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Time = 0.22 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {52, 65, 231} \[ \int \sqrt {a+b x} \sqrt [6]{c+d x} \, dx=-\frac {3\ 3^{3/4} \sqrt [6]{c+d x} (b c-a d)^{5/3} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{40 b d^2 \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac {3 \sqrt {a+b x} \sqrt [6]{c+d x} (b c-a d)}{20 b d}+\frac {3 (a+b x)^{3/2} \sqrt [6]{c+d x}}{5 b} \]
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Rule 52
Rule 65
Rule 231
Rubi steps \begin{align*} \text {integral}& = \frac {3 (a+b x)^{3/2} \sqrt [6]{c+d x}}{5 b}+\frac {(b c-a d) \int \frac {\sqrt {a+b x}}{(c+d x)^{5/6}} \, dx}{10 b} \\ & = \frac {3 (b c-a d) \sqrt {a+b x} \sqrt [6]{c+d x}}{20 b d}+\frac {3 (a+b x)^{3/2} \sqrt [6]{c+d x}}{5 b}-\frac {\left (3 (b c-a d)^2\right ) \int \frac {1}{\sqrt {a+b x} (c+d x)^{5/6}} \, dx}{40 b d} \\ & = \frac {3 (b c-a d) \sqrt {a+b x} \sqrt [6]{c+d x}}{20 b d}+\frac {3 (a+b x)^{3/2} \sqrt [6]{c+d x}}{5 b}-\frac {\left (9 (b c-a d)^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-\frac {b c}{d}+\frac {b x^6}{d}}} \, dx,x,\sqrt [6]{c+d x}\right )}{20 b d^2} \\ & = \frac {3 (b c-a d) \sqrt {a+b x} \sqrt [6]{c+d x}}{20 b d}+\frac {3 (a+b x)^{3/2} \sqrt [6]{c+d x}}{5 b}-\frac {3\ 3^{3/4} (b c-a d)^{5/3} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{b c-a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\cos ^{-1}\left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{40 b d^2 \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.18 \[ \int \sqrt {a+b x} \sqrt [6]{c+d x} \, dx=\frac {2 (a+b x)^{3/2} \sqrt [6]{c+d x} \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {3}{2},\frac {5}{2},\frac {d (a+b x)}{-b c+a d}\right )}{3 b \sqrt [6]{\frac {b (c+d x)}{b c-a d}}} \]
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\[\int \sqrt {b x +a}\, \left (d x +c \right )^{\frac {1}{6}}d x\]
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\[ \int \sqrt {a+b x} \sqrt [6]{c+d x} \, dx=\int { \sqrt {b x + a} {\left (d x + c\right )}^{\frac {1}{6}} \,d x } \]
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\[ \int \sqrt {a+b x} \sqrt [6]{c+d x} \, dx=\int \sqrt {a + b x} \sqrt [6]{c + d x}\, dx \]
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\[ \int \sqrt {a+b x} \sqrt [6]{c+d x} \, dx=\int { \sqrt {b x + a} {\left (d x + c\right )}^{\frac {1}{6}} \,d x } \]
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\[ \int \sqrt {a+b x} \sqrt [6]{c+d x} \, dx=\int { \sqrt {b x + a} {\left (d x + c\right )}^{\frac {1}{6}} \,d x } \]
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Timed out. \[ \int \sqrt {a+b x} \sqrt [6]{c+d x} \, dx=\int \sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{1/6} \,d x \]
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